The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

Order Stars and a Saturation Theorem for First-order Hyperbolics

We prove that stable numerical finite difference methods forfirst-order hyperbolics, which use s forward and r backwardsteps in the discretization of the space derivatives, are oforder at most 2 min{r+1, s}. This generalizes results of Strang(1964) and of Engquist & Osher (1980b). We also derive linearstability results for interpolatory finite differences. Thegiven analysis is based on a generalization of the theory oforder stars.     

Abstract Cauchy Problems for Quasi-Linear Evolution Equations in the Sense of Hadamard

This paper is devoted to the well-posedness of abstract Cauchyproblems for quasi-linear evolution equations. The notion ofHadamard well-posedness is considered, and a new type of stabilitycondition is introduced from the viewpoint of the theory offinite difference approximations. The result obtained here generalizesnot only some results on abstract Cauchy problems closely relatedwith the theory of integrated semigroups or regularized semigroupsbut also the Kato theorem on quasi-linear evolution equations.An application to some quasi-linear partial differential equationof weakly hyperbolic type is also given. 2000 Mathematics SubjectClassification 34G20, 47J25 (primary), 47D60, 47D62 (secondary).     

Initial Development of Diffusion in Poiseuille Flow

The combined effect of diffusion, and of convection by Poiseuilleflow, on the distribution of a small quantity of miscible additiveinjected into a tube of radius a, is to spread it longitudinallywith a Taylor "effective diffusion coefficient", to an approximationthat is good at times greater than about 0.5a²/D (Bailey &Gogarty, 1962), where D is the molecular diffusion coefficient.The present theory, complementary to the Taylor theory, determinesthe initial action of diffusion on the front of the concentrationdistribution, to an approximation that is good at times t lessthan about 0.1a²/D. The theory is exact wherever the added substancedoes not yet interact with the tube wall, and predicts thatthe spread in the front due to diffusion extends (Fig. 2) overa distance of order DUt²/a², where U is the velocity on theaxis of the tube. The transition between distributions characteristicof the two theories is illustrated (Fig. 4); and the introductionindicates the relevance of the new theory to work (Caro, 1966)on tracers used in study of the blood circulation.     

On the Norm Equation Over Function Fields

If K is an algebraic function field of one variable over analgebraically closed field k and F is a finite extension ofK, then any element a of K can be written as a norm of someb in F by Tsen's theorem. All zeros and poles of a lead to zerosand poles of b, but in general additional zeros and poles occur.The paper shows how this number of additional zeros and polesof b can be restricted in terms of the genus of K, respectivelyF. If k is the field of all complex numbers, then we use Abel'stheorem concerning the existence of meromorphic functions ona compact Riemann surface. From this, the general case of characteristic0 can be derived by means of principles from model theory, sincethe theory of algebraically closed fields is model-complete.Some of these results also carry over to the case of characteristicp>0 using standard arguments from valuation theory.     

Finite-order meromorphic solutions and the discrete Painleve equations

Let w(z) be an admissible finite-order meromorphic solutionof the second-order difference equation

where R(z, w(z)) is rational in w(z) withcoefficients that are meromorphic in z. Then either w(z) satisfiesa difference linear or Riccati equation or else the above equationcan be transformed to one of a list of canonical differenceequations. This list consists of all known difference Painlevéequations of the above form, together with their autonomousversions. This suggests that the existence of finite-order meromorphicsolutions is a good detector of integrable difference equations.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

Constructions of Partial Difference Sets and Relative Difference Sets on p-Groups

By using some finite local rings, we construct some new partialdifference sets and relative difference sets on pgroups wherep is any prime. When p = 2, some of partial difference setsconstructed are reversible difference sets which include Dillon'sdifference sets.     

L1 Properties of Second order Elliptic Operators

We present a connected account of various spectral and regularityproperties of the semigroup associated with a fairly generalsymmetric second order elliptic operator on a Riemannian manifold.Our main goal is to relate the L² theory to the less well understoodL¹ theory, and hence to the approach via the theory of stochasticdifferential equations.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.     

On Symmetric Cubature Formulae for Planar Regions

This paper is concerned with , D_m-symmetric, cubature formulaefor D_m-symmetric planar regions of integration, D_m being thedihedral group of order 2m, that is, the symmetry group of aregular polygon with m edges. A unified theory for the analysisof this kind of formula set is introduced. This theory arisesfrom the identification of the space of all polynomials thatare invariant with respect to the symmetry group D_m. The typeof analysis used leads to a simple method for constructing thiskind of cubature formula set, even when a high degree of polynomialprecision is required.     

On the Structure of Alternating Direction Implicit (A.D.I.) and Locally One Dimensional (L.O.D.) Difference Methods

The general structure of A.D.I. and L.O.D. difference schemesis considered with regard to their construction for time dependentproblems in two and three space dimensions. By considering approximationsto exp {k(L+M)} where L and M are differential operators inthe space variables and k is the time step, we show how severalknown schemes can be viewed as having come from this type ofapproximation. In addition several new schemes based on thistype of approximation are suggested. The arguments used areentirely informal and no attempt is made to prove the stabilityor convergence of the various schemes. Our aim is merely topoint out a possible structure for the generation of A.D.I.and L.O.D. difference schemes.     

A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbortransition probability that yields a recurrent random walk andshow that, although such trees have no positive potentials,many of the standard results of potential theory can be transferredto this setting. We accomplish this by defining a non-negativefunction H, harmonic outside the root e and vanishing only ate, and a substitute notion of potential which we call H-potential.We define the flux of a superharmonic function outside a finiteset of vertices, give some simple formulas for calculating theflux and derive a global Riesz decomposition theorem for superharmonicfunctions with a harmonic minorant outside a finite set. Wediscuss the connection of the H-potentials with other notionsof potentials for recurrent Markov chains in the literature.     

L-splines--a manifestation of optimal control

We show connections between L-splines and optimal control theory,leading to the conclusion that L-splines are manifestationsof an optimal behaviour.     

A mathematical model of silicon-based thermobiosensors

A mathematical model of a silicon glucose thermobiosensor thatdetects changes in temperature produced by a biocatalytic reactionis proposed for calculation of transient temperature and reactantconcentration profiles, time-dependencies of the output signal,and calibration curves. Mathematically the model is reducedto a one-dimensional linear initial-boundary-value problem ofthe heat-conduction equation with a thermal source F(x, t).In order to find F(x, t), a system of the second-order nonlinearpartial differential equations for glucose and oxygen concentrations,describing a combination of diffusion-membrane theory and Michaelis-Mentenenzyme reaction theory, has been solved. The computed dependenciesof transient temperature profile and sensor response to variousconditions such as oxygen buffer concentration, membrane thickness,enzyme loading, and operation mode are analysed for the optimaldesign of the tensor.     

Asymptotic approximations for functions defined by series, with some applications to the theory of guided waves

Mellin transforms are used here to find asymptotic approximationsfor functions defined by series. The simplest cases are thoseof the form _–1;u(nx). Such series are called separablehere, because the given function u is sampled at points whosevariation with n and x is separated. Nonseparable series areanalysed by first approximating them by separable series. Bothtypes of series arise in the theory of electromagnetic waveguidesand in the theory of linear water waves; several examples areworked out in detail.     

Plain Representations of Lie Algebras

In this paper we study representations of finite dimensionalLie algebras. In this case representations are not necessarilycompletely reducible. As the general problem is known to beof enormous complexity, we restrict ourselves to representationsthat behave particularly well on Levi subalgebras. We call suchrepresentations plain (Definition 1.1). Informally, we showthat the theory of plain representations of a given Lie algebraL is equivalent to representation theory of finitely many finitedimensional associative algebras, also non-semisimple. The senseof this is to distinguish representations of Lie algebras thatare of complexity comparable with that of representations ofassociative algebras. Non-plain representations are intrinsicallymuch more complex than plain ones. We view our work as a steptoward understanding this complexity phenomenon. We restrict ourselves also to perfect Lie algebras L, that is,such that L = [L, L]. In our main results we assume that L isperfect and sl₂-free (which means that L has no quotient isomorphicto sl₂). The ground field F is always assumed to be algebraicallyclosed and of characteristic 0.     

Elliptic and Hyperelliptic Curves Over Supersimple Fields

It is proved that if F is an infinite field with characteristicdifferent from 2, whose theory is supersimple, and C is an ellipticor hyperelliptic curve over F with generic ‘modulus’,then C has a generic F-rational point. The notion of generityhere is in the sense of the supersimple field F.     

From Endomorphisms to Automorphisms and Back: Dilations and Full Corners

When S is a discrete subsemigroup of a discrete group G suchthat G = S^–1S, it is possible to extend circle-valuedmultipliers from S to G, to dilate (projective) isometric representationsof S to (projective) unitary representations of G, and to dilate/extendactions of S by injective endomorphisms of a C*-algebra to actionsof G by automorphisms of a larger C*-algebra. These dilationsare unique provided they satisfy a minimality condition. The(twisted) semigroup crossed product corresponding to an actionof S is isomorphic to a full corner in the (twisted) crossedproduct by the dilated action of G. This shows that crossedproducts by semigroup actions are Morita equivalent to crossedproducts by group actions, making powerful tools available tostudy their ideal structure and representation theory. The dilationof the system giving the Bost–Connes Hecke C*-algebrafrom number theory is constructed explicitly as an application:it is the crossed product C₀(A_f)Q*₊, corresponding to the multiplicativeaction of the positive rationals on the additive group A_f offinite adeles.     

Homogenization of the Stefan Problem and Application to Magnetic Composite Media

The theory of homogenization (Bensoussan, Lions & Papanicolaou,1978) shows that u, the solution of the diffusion equation [with k(y) periodic in the space-variable y and q = cu a linearfunction of u] has a weak limit u for = 0. This theory allowsone to compute, for a given k, the conductivity tensor of ananisotropic but homogeneous medium in which, for unchanged initialand boundary conditions, u is the solution of the diffusionequation. We examine here the case where the relation between q and uis given by a maximal monotone graph (i.e. the Stefan problem),depending on the space variable in the same manner as k. Applicationsto eddy-current problems in magnetic composite media (steelcables, laminations) are suggested. A numerical example is given.